Theorem subgex 13249

Description: The class of subgroups of a group is a set. (Contributed by Jim Kingdon, 8-Mar-2025.)

Assertion

Ref	Expression
subgex	|- ( G e. Grp -> (SubGrp ` G ) e. _V )

Proof of Theorem subgex

Dummy variables s w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-subg 13243	. . 3 |- SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
2		fveq2 5555	. . . . 5 |- ( w = G -> ( Base ` w ) = ( Base ` G ) )
3	2	pweqd 3607	. . . 4 |- ( w = G -> ~P ( Base ` w ) = ~P ( Base ` G ) )
4		oveq1 5926	. . . . 5 |- ( w = G -> ( w ↾s s ) = ( G ↾s s ) )
5	4	eleq1d 2262	. . . 4 |- ( w = G -> ( ( w ↾s s ) e. Grp <-> ( G ↾s s ) e. Grp ) )
6	3, 5	rabeqbidv 2755	. . 3 |- ( w = G -> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } = { s e. ~P ( Base ` G ) | ( G ↾s s ) e. Grp } )
7		id 19	. . 3 |- ( G e. Grp -> G e. Grp )
8		basfn 12679	. . . . . 6 |- Base Fn _V
9		elex 2771	. . . . . 6 |- ( G e. Grp -> G e. _V )
10		funfvex 5572	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
11	10	funfni 5355	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
12	8, 9, 11	sylancr 414	. . . . 5 |- ( G e. Grp -> ( Base ` G ) e. _V )
13	12	pwexd 4211	. . . 4 |- ( G e. Grp -> ~P ( Base ` G ) e. _V )
14		rabexg 4173	. . . 4 |- ( ~P ( Base ` G ) e. _V -> { s e. ~P ( Base ` G ) | ( G ↾s s ) e. Grp } e. _V )
15	13, 14	syl 14	. . 3 |- ( G e. Grp -> { s e. ~P ( Base ` G ) | ( G ↾s s ) e. Grp } e. _V )
16	1, 6, 7, 15	fvmptd3 5652	. 2 |- ( G e. Grp -> (SubGrp ` G ) = { s e. ~P ( Base ` G ) | ( G ↾s s ) e. Grp } )
17	16, 15	eqeltrd 2270	1 |- ( G e. Grp -> (SubGrp ` G ) e. _V )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   {crab 2476   _Vcvv 2760   ~Pcpw 3602   Fn wfn 5250   ` cfv 5255  (class class class)co 5919   Basecbs 12621   ↾_s cress 12622   Grpcgrp 13075  SubGrpcsubg 13240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5922  df-inn 8985  df-ndx 12624  df-slot 12625  df-base 12627  df-subg 13243

This theorem is referenced by:  isnsg  13275